An introduction to Quantum computing

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Transcript of An introduction to Quantum computing

An introduction to Quantum computingQuantum computer hardwareGot bits ? need gatesApplied computer scienceInteger factorization (Peter Shor)Quantum bitsMeet a QubitTheoretical computer scienceDatabase search (Lov Grover)Hmm.. What the hell?quick definitionoriginshopespresentation scopewhy bother ?Telecommunication securityHow to harness its powerWhat I could computeWhat I can computeWhat's next ?Quantum mechanicsTiny weird objectsA new securityparadigmFresh new bites !!!Turing and complexity theory010 101referencescedric codet - 07/11/2013 - Jeudi TrialogA quantum computer is a machine that performs calculations based on the laws of quantum mechanics, which is the behavior of particles at the sub-atomic level.1982 - Feynman proposed the idea of creating machines based on the laws of quantum mechanics instead of the laws of classical physics.1985 - David Deutsch developed the quantum turing machine, showing that quantum circuits are universal.1994 - Peter Shor came up with a quantum algorithm to factor very large numbers in polynomial time.1997 - Lov Grover develops a quantum search algorithm with O(√N) complexitydifferent kinds of QubitsDue to the nature of quantum physics, the destruction of information in a gate will cause heat to be evolved which can destroy the superposition of qubits.This type of gate cannot be used. We must use Quantum Gates.Quantum Gates are similar to classical gates, but do not have a degenerate output. i.e. their original input state can be derived from their output state, uniquely. They must be reversible.

This means that a deterministic computation can be performed on a quantum computer only if it is reversible. Luckily, it has been shown that any deterministic computation can be made reversible.(Charles Bennet, 1973)Simplest gate involves one qubit and is called a Hadamard Gate (also known as a square-root of NOT gate.) Used to put qubits into superposition.more stuff... AndThe CCN gate has been shown to be a universal reversible logic gate as it can be used as a NAND gate.Application: cracking RSA !if you can factorize (quick enough) the public key, you can decipher the messageexisting computersIBM quantum molecule computerD-Wave- Key technical challenge: prevent decoherence, or unwanted interaction with environment- Several Qubits approaches: NMR, ion trap, quantum dot, Josephson junction, optical…implementation- Larger computations will require quantum error- correcting codesExpected applications- Quantum computing can provide insight on the physical nature of the world- On the other hand, physical processes may enable to solve computing problems (like optimizations)- Secured commmunications from eavesdropper- Crypto analysis- Faster search algorithms- Quantum systems simulationNeed for increase in processing power strong (big data processing has become a necessity for businesses)

Moore’s Law: We hit the quantum level by 2010~2020. “No, you’re not going to be able to understand it. . . . You see, my physics students don’t understand it either. That is because I don’t understand it. Nobody does. ... The theory of quantum electrodynamics describes Nature as absurd from the point of view of common sense. And it agrees fully with an experiment. So I hope that you can accept Nature as She is -- absurd.

Richard FeynmanThe Photon caseA simple experiment with polarization"Explanation"What's a Photon ?Polarization ?The experimentVibration of photon|phi> = a|↑> + b|→>probability that the photon polarized is horizontally (switch to state |→>) after measure: |b|² |a²|+|b²| = 1Duality body wave:- To explain some aspects of light behavior, such as interference and diffraction, you treat it as a wave- To explain other aspects you treat light as being made up of particlesprobability that the photon is polarized vertically (switch to state |↑>) after measure: |a|² a and b are complex numbersthe basis is arbitraryKey notionsentanglementno cloning destructive measureprobabilistic behavioursuper coolsuckssuperposition (= coherence)decoherenceSuper computerThe photon as a Qubit|phi> = a|↑> + b|→>can be written|phi> = a|1> + b|0>quantum object used to store information: that's a Qubit !Quantum Key DistributionMultiple QubitsSafe communication channel-Classical bits : independent0 1those two bits have no interaction between themQubits: entangled2 parameters are sufficient to describe the state of the register=> 2 classical bits carry 2 unity of informationbit1bit2bit1bit2Decoherencea|1> + b|0>a'|1> + b'|0>NO!Experience proves qubits can't be described independently. there is a strong correlation between states.you need a new basis to describe the two bitsEPR pair and entanglement|bit1 & bit2> = w|00> + x|10> + y|01> + z|11>What can you do ? => you code all integers up to 2^n-1 on n bits AT THE SAME TIME

If you have a 2 qubits computer => you can store the following integers:- 0 coded by the physical state |00>- 1 coded by |01>- 2 coded by |10>- 3 coded by |11>and compute all of them at the same time|00>|01>|10>|11>bit1bit2No cloning principlemiddle man attack hard because no cloning principle-4 parameters are need to describe the state of the register=> 2 qubits carry 4 unity of informationexponential growth of the state spaceMassive code parallelization ?No : Destructive measure + probabilitic output => you got only one result! Maths say so! The proof relies on Quantum mechanism axioms and particularly linearity.Any function modifying a state |phi> must be linear. You can't find such a function that copies the state of a qubit to another.You could copy all the results before measuring ? No: No cloning theoremYou have to invent another way to code => Software challenge !Computation:- add bit1 and bit2- write the result in bit3

"Decoherence, in brief, describes the constant, tenuous interactions between a system or object and its environment, a set of interactions that allows concrete behaviors to emerge from the multitude of simultaneous possibilities that quantum theory allows".

Schrodinger's cat is not in a superposition of states. The number of interacting particles "stabilizes" it all.

To be done !Gate examplehttp://imgur.com/gallery/a4yU9http://en.wikipedia.org/wiki/Timeline_of_quantum_computing- The process has to be repeated several times because the QFT may yield false results- Most quantum algorithms : they are probabilistic and give the right answer with high probabilityHow to find the period of a function f on a quantum computer ? Input registerOutput registerEntangled QubitsInitialize registerssuperposition of integers from 0 to Log2(N). All Qubits set to 0Apply f to input registerN Qubitssuperposition of integers from 0 to Log2(N). superposition of f(n) with n integer from 0 to Log2(N). n and f(n) are entangledWhen measuring f(x), the input register collapses to the superposition of states y such that f(y) = f(x) Measure f(x)superposition of states x and y such that f(y) = f(x) f(x) f(x)x0123|00>|01>|10>|11>input registeroutput register01|1>|0>if we measure 1:f(x)x13|01>|11>input registeroutput register01|1>|0>Apply QFT on input registerBiggest difficulty: decoherenceNeed for very low temperature, magnetic fields...superposition of states x and y such that f(y) = f(x) The period of f !- Performed factorization of 15 with shor's algorithmClassical computing poses no serious threat to the RSA system- approximately 428 millenia to factorise a 200-digit number (assuming GHz computing speed).- Shor’s quantum algorithm would be able to factor the same number in a matter of daysGrover's algorithm is a quantum algorithm for searching an unsorted database with N entries in O(N1/2) time and using O(log N) storage space. Lov Grover formulated it in 1996.In models of classical computation, searching an unsorted database cannot be done in less than linear time (so merely searching through every item is optimal). Grover's algorithm illustrates that in the quantum model searching can be done faster than this; in fact its time complexity O(N1/2) is asymptotically the fastest possible for searching an unsorted database in the linear quantum model.Can find a solution in polynomial time (bounded by N^x)N is the size of the inputdifficulty of a problemCan verify a solution in polynomial time (bounded by N^x)Requires a space on a computer bounded by N^xBounded-Error Probabilistic Polynomial-Time with error probability under 1/3Bounded error Quantum Polynomial time with error probability under 1/3Bounded-Error Probabilistic Polynomial-Time with error probability under 1/2Prospects- Bought by Lockhead Martin, Google and Nasa- Big controversy: is it a quantum computer ?- In parallel, new programming techniques are required- Link between computability and physicsHectic, hard to follow =>- Compute hard problems- Harness quantum mechanics properties for security purposes- Deepen our understanding of PhysicsBeware !-> More a mathematical theory than a physical onePolarization in photographyA basis for QubitsComputation exampleOutline of the algorithmQuantum partRemarks